List of Johnson solids

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In geometry, a convex polyhedron whose faces are regular polygons is known as a Johnson solid, or sometimes as a Johnson–Zalgaller solid.^[1] Some authors exclude uniform polyhedra (in which all vertices are symmetric to each other) from the definition; uniform polyhedra include Platonic and Archimedean solids as well as prisms and antiprisms.^[2] The Johnson solids are named after American mathematician Norman Johnson (1930–2017), who published a list of 92 non-uniform Johnson polyhedra in 1966. His conjecture that the list was complete and no other examples existed was proven by Russian-Israeli mathematician Victor Zalgaller (1920–2020) in 1969.^[3]

Some of the Johnson solids may be categorized as elementary polyhedra, meaning they cannot be separated by a plane to create two small convex polyhedra with regular faces. The Johnson solids satisfying this criteria are the first six—equilateral square pyramid, pentagonal pyramid, triangular cupola, square cupola, pentagonal cupola, and pentagonal rotunda. The criteria is also satisfied by eleven other Johnson solids, specifically the tridiminished icosahedron, parabidiminished rhombicosidodecahedron, tridiminished rhombicosidodecahedron, snub disphenoid, snub square antiprism, sphenocorona, sphenomegacorona, hebesphenomegacorona, disphenocingulum, bilunabirotunda, and triangular hebesphenorotunda.^[4] The rest of the Johnson solids are not elementary, and they are constructed using the first six Johnson solids together with Platonic and Archimedean solids in various processes. Augmentation involves attaching the Johnson solids onto one or more faces of polyhedra, while elongation or gyroelongation involve joining them onto the bases of a prism or antiprism, respectively. Some others are constructed by diminishment, the removal of one of the first six solids from one or more of a polyhedron's faces.^[5]

The following table contains the 92 Johnson solids, with edge length $a$ . The table includes the solid's enumeration (denoted as $J_{n}$ ).Template:Sfnp It also includes the number of vertices, edges, and faces of each solid, as well as its symmetry group, surface area $A$ , and volume $V$ . Every polyhedron has its own characteristics, including symmetry and measurement. An object is said to have symmetry if there is a transformation that maps it to itself. All of those transformations may be composed in a group, alongside the group's number of elements, known as the order. In two-dimensional space, these transformations include rotating around the center of a polygon and reflecting an object around the perpendicular bisector of a polygon. A polygon that is rotated symmetrically by $\frac{36 0^{\circ}}{n}$ is denoted by $C_{n}$ , a cyclic group of order $n$ ; combining this with the reflection symmetry results in the symmetry of dihedral group $D_{n}$ of order $2 n$ .^[6] In three-dimensional symmetry point groups, the transformations preserving a polyhedron's symmetry include the rotation around the line passing through the base center, known as the axis of symmetry, and the reflection relative to perpendicular planes passing through the bisector of a base, which is known as the pyramidal symmetry $C_{n v}$ of order $2 n$ . The transformation that preserves a polyhedron's symmetry by reflecting it across a horizontal plane is known as the prismatic symmetry $D_{n h}$ of order $4 n$ . The antiprismatic symmetry $D_{n d}$ of order $4 n$ preserves the symmetry by rotating its half bottom and reflection across the horizontal plane.Template:Sfnp The symmetry group $C_{n h}$ of order $2 n$ preserves the symmetry by rotation around the axis of symmetry and reflection on the horizontal plane; the specific case preserving the symmetry by one full rotation is $C_{1 h}$ of order 2, often denoted as $C_{s}$ .^[7] The mensuration of polyhedra includes the surface area and volume. An area is a two-dimensional measurement calculated by the product of length and width; for a polyhedron, the surface area is the sum of the areas of all of its faces.Template:Sfnp A volume is a measurement of a region in three-dimensional space.Template:Sfnp The volume of a polyhedron may be ascertained in different ways: either through its base and height (like for pyramids and prisms), by slicing it off into pieces and summing their individual volumes, or by finding the root of a polynomial representing the polyhedron.^[8] Template:Mw-datatable Template:Sticky-header Template:Sort-under

The 92 Johnson solids
$J_{n}$	Solid name	Vertices	Edges	Faces	Symmetry group and its order Template:Sfnp	Surface area and volumeTemplate:Sfnp
1	Equilateral square pyramid	5	8	5	$C_{4 v}$ of order 8	$\begin{matrix} A & = (1 + \sqrt{3}) a^{2} \\ \approx 2.7321 a^{2} \\ V & = \frac{\sqrt{2}}{6} a^{3} \\ \approx 0.2357 a^{3} \end{matrix}$
2	Pentagonal pyramid	6	10	6	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{a^{2}}{2} \sqrt{\frac{5}{2} (10 + \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})} \\ \approx 3.8855 a^{2} \\ V & = (\frac{5 + \sqrt{5}}{24}) a^{3} \\ \approx 0.3015 a^{3} \end{matrix}$
3	Triangular cupola	9	15	8	$C_{3 v}$ of order 6	$\begin{matrix} A & = (3 + \frac{5 \sqrt{3}}{2}) a^{2} \\ \approx 7.3301 a^{2} \\ V & = (\frac{5}{3 \sqrt{2}}) a^{3} \\ \approx 1.1785 a^{3} \end{matrix}$
4	Square cupola	12	20	10	$C_{4 v}$ of order 8	$\begin{matrix} A & = (7 + 2 \sqrt{2} + \sqrt{3}) a^{2} \\ \approx 11.5605 a^{2} \\ V & = (1 + \frac{2 \sqrt{2}}{3}) a^{3} \\ \approx 1.9428 a^{3} \end{matrix}$
5	Pentagonal cupola	15	25	12	$C_{5 v}$ of order 10	$\begin{matrix} A & = (\frac{1}{4} (20 + 5 \sqrt{3} + \sqrt{5 (145 + 62 \sqrt{5})})) a^{2} \\ \approx 16.5798 a^{2} \\ V & = (\frac{1}{6} (5 + 4 \sqrt{5})) a^{3} \\ \approx 2.3241 a^{3} \end{matrix}$
6	Pentagonal rotunda	20	35	17	$C_{5 v}$ of order 10	$\begin{matrix} A & = (\frac{1}{2} (5 \sqrt{3} + \sqrt{10 (65 + 29 \sqrt{5})})) a^{2} \\ \approx 22.3472 a^{2} \\ V & = (\frac{1}{12} (45 + 17 \sqrt{5})) a^{3} \\ \approx 6.9178 a^{3} \end{matrix}$
7	Elongated triangular pyramid	7	12	7	$C_{3 v}$ of order 6	$\begin{matrix} A & = (3 + \sqrt{3}) a^{2} \\ \approx 4.7321 a^{2} \\ V & = (\frac{1}{12} (\sqrt{2} + 3 \sqrt{3})) a^{3} \\ \approx 0.5509 a^{3} \end{matrix}$
8	Elongated square pyramid	9	16	9	$C_{4 v}$ of order 8	$\begin{matrix} A & = (5 + \sqrt{3}) a^{2} \\ \approx 6.7321 a^{2} \\ V & = (1 + \frac{\sqrt{2}}{6}) a^{3} \\ \approx 1.2357 a^{3} \end{matrix}$
9	Elongated pentagonal pyramid	11	20	11	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{20 + 5 \sqrt{3} + \sqrt{25 + 10 \sqrt{5}}}{4} a^{2} \\ \approx 8.8855 a^{2} \\ V & = (\frac{5 + \sqrt{5} + 6 \sqrt{25 + 10 \sqrt{5}}}{24}) a^{3} \\ \approx 2.022 a^{3} \end{matrix}$
10	Gyroelongated square pyramid	9	20	13	$C_{4 v}$ of order 8	$\begin{matrix} A & = (1 + 3 \sqrt{3}) a^{2} \\ \approx 6.1962 a^{2} \\ V & = \frac{1}{6} (\sqrt{2} + 2 \sqrt{4 + 3 \sqrt{2}}) a^{3} \\ \approx 1.1927 a^{3} \end{matrix}$
11	Gyroelongated pentagonal pyramid	11	25	16	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (15 \sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 8.2157 a^{2} \\ V & = \frac{1}{24} (25 + 9 \sqrt{5}) a^{3} \\ \approx 1.8802 a^{3} \end{matrix}$
12	Triangular bipyramid	5	9	6	$D_{3 h}$ of order 12	$\begin{matrix} A & = \frac{3 \sqrt{3}}{2} a^{2} \\ \approx 2.5981 a^{2} \\ V & = \frac{\sqrt{2}}{6} a^{3} \\ \approx 0.2358 a^{3} \end{matrix}$
13	Pentagonal bipyramid	7	15	10	$D_{5 h}$ of order 20	$\begin{matrix} A & = \frac{5 \sqrt{3}}{2} a^{2} \\ \approx 4.3301 a^{2} \\ V & = \frac{1}{12} (5 + \sqrt{5}) a^{3} \\ \approx 0.603 a^{3} \end{matrix}$
14	Elongated triangular bipyramid	8	15	9	$D_{3 h}$ of order 12	$\begin{matrix} A & = \frac{3}{2} (2 + \sqrt{3}) a^{2} \\ \approx 5.5981 a^{2} \\ V & = \frac{1}{12} (2 \sqrt{2} + 3 \sqrt{3}) a^{3} \\ \approx 0.6687 a^{3} \end{matrix}$
15	Elongated square bipyramid	10	20	12	$D_{4 h}$ of order 16	$\begin{matrix} A & = 2 (2 + \sqrt{3}) a^{2} \\ \approx 7.4641 a^{2} \\ V & = \frac{1}{3} (3 + \sqrt{2}) a^{3} \\ \approx 1.4714 a^{3} \end{matrix}$
16	Elongated pentagonal bipyramid	12	25	15	$D_{5 h}$ of order 20	$\begin{matrix} A & = \frac{5}{2} (2 + \sqrt{3}) a^{2} \\ \approx 9.3301 a^{2} \\ V & = \frac{1}{12} (5 + \sqrt{5} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{3} \\ \approx 2.3235 a^{3} \end{matrix}$
17	Gyroelongated square bipyramid	10	24	16	$D_{4 d}$ of order 16	$\begin{matrix} A & = 4 \sqrt{3} a^{2} \\ \approx 6.9282 a^{2} \\ V & = \frac{1}{3} (\sqrt{2} + \sqrt{4 + 3 \sqrt{2}}) a^{3} \\ \approx 1.4284 a^{3} \end{matrix}$
18	Elongated triangular cupola	15	27	14	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{2} (18 + 5 \sqrt{3}) a^{2} \\ \approx 13.3301 a^{2} \\ V & = \frac{1}{6} (5 \sqrt{2} + 9 \sqrt{3}) a^{3} \\ \approx 3.7766 a^{3} \end{matrix}$
19	Elongated square cupola	20	36	18	$C_{4 v}$ of order 8	$\begin{matrix} A & = (15 + 2 \sqrt{2} + \sqrt{3}) a^{2} \\ \approx 19.5605 a^{2} \\ V & = (3 + \frac{8 \sqrt{2}}{3}) a^{3} \\ \approx 6.7712 a^{3} \end{matrix}$
20	Elongated pentagonal cupola	25	45	22	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (60 + 5 \sqrt{3} + 10 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 26.5798 a^{2} \\ V & = \frac{1}{6} (5 + 4 \sqrt{5} + 15 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 10.0183 a^{3} \end{matrix}$
21	Elongated pentagonal rotunda	30	55	27	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{2} a^{2} (20 + 5 \sqrt{3} + 5 \sqrt{5 + 2 \sqrt{5}} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) \\ \approx 32.3472 a^{2} \\ V & = \frac{1}{12} a^{3} (45 + 17 \sqrt{5} + 30 \sqrt{5 + 2 \sqrt{5}}) \\ \approx 14.612 a^{3} \end{matrix}$
22	Gyroelongated triangular cupola	15	33	20	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{2} (6 + 11 \sqrt{3}) a^{2} \\ \approx 12.5263 a^{2} \\ V & = \frac{1}{3} \sqrt{\frac{61}{2} + 18 \sqrt{3} + 30 \sqrt{1 + \sqrt{3}}} a^{3} \\ \approx 3.5161 a^{3} \end{matrix}$
23	Gyroelongated square cupola	20	44	26	$C_{4 v}$ of order 8	$\begin{matrix} A & = (7 + 2 \sqrt{2} + 5 \sqrt{3}) a^{2} \\ \approx 18.4887 a^{2} \\ V & = (1 + \frac{2}{3} \sqrt{2} + \frac{2}{3} \sqrt{4 + 2 \sqrt{2} + 2 \sqrt{146 + 103 \sqrt{2}}}) a^{3} \\ \approx 6.2108 a^{3} \end{matrix}$
24	Gyroelongated pentagonal cupola	25	55	32	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (20 + 25 \sqrt{3} + 10 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 25.2400 a^{2} \\ V & = (\frac{5}{6} + \frac{2}{3} \sqrt{5} + \frac{5}{6} \sqrt{2 \sqrt{650 + 290 \sqrt{5}} - 2 \sqrt{5} - 2}) a^{3} \\ \approx 9.0733 a^{3} \end{matrix}$
25	Gyroelongated pentagonal rotunda	30	65	37	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{2} (15 \sqrt{3} + (5 + 3 \sqrt{5}) \sqrt{5 + 2 \sqrt{5}}) a^{2} \\ \approx 31.0075 a^{2} \\ V & = (\frac{45}{12} + \frac{17}{12} \sqrt{5} + \frac{5}{6} \sqrt{2 \sqrt{650 + 290 \sqrt{5}} - 2 \sqrt{5} - 2}) a^{3} \\ \approx 13.6671 a^{3} \end{matrix}$
26	Gyrobifastigium	8	14	8	$D_{2 d}$ of order 8	$\begin{matrix} A & = (4 + \sqrt{3}) a^{2} \\ \approx 5.7321 a^{2} \\ V & = (\frac{\sqrt{3}}{2}) a^{3} \\ \approx 0.866 a^{3} \end{matrix}$
27	Triangular orthobicupola	12	24	14	$D_{3 h}$ of order 12	$\begin{matrix} A & = 2 (3 + \sqrt{3}) a^{2} \\ \approx 9.4641 a^{2} \\ V & = \frac{5 \sqrt{2}}{3} a^{3} \\ \approx 2.357 a^{3} \end{matrix}$
28	Square orthobicupola	16	32	18	$D_{4 h}$ of order 16	$\begin{matrix} A & = 2 (5 + \sqrt{3}) a^{2} \\ \approx 13.4641 a^{2} \\ V & = (2 + \frac{4 \sqrt{2}}{3}) a^{3} \\ \approx 3.8856 a^{3} \end{matrix}$
29	Square gyrobicupola	16	32	18	$D_{4 d}$ of order 16	$\begin{matrix} A & = 2 (5 + \sqrt{3}) a^{2} \\ \approx 13.4641 a^{2} \\ V & = (2 + \frac{4 \sqrt{2}}{3}) a^{3} \\ \approx 3.8856 a^{3} \end{matrix}$
30	Pentagonal orthobicupola	20	40	22	$D_{5 h}$ of order 20	$\begin{matrix} A & = (10 + \sqrt{\frac{5}{2} (10 + \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 17.7711 a^{2} \\ V & = \frac{1}{3} (5 + 4 \sqrt{5}) a^{3} \\ \approx 4.6481 a^{3} \end{matrix}$
31	Pentagonal gyrobicupola	20	40	22	$D_{5 d}$ of order 20	$\begin{matrix} A & = (10 + \sqrt{\frac{5}{2} (10 + \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 17.7711 a^{2} \\ V & = \frac{1}{3} (5 + 4 \sqrt{5}) a^{3} \\ \approx 4.6481 a^{3} \end{matrix}$
32	Pentagonal orthocupolarotunda	25	50	27	$C_{5 v}$ of order 10	$\begin{matrix} A & = (5 + \frac{1}{4} \sqrt{1900 + 490 \sqrt{5} + 210 \sqrt{75 + 30 \sqrt{5}}}) a^{2} \\ \approx 23.5385 a^{2} \\ V & = \frac{5}{12} (11 + 5 \sqrt{5}) a^{3} \\ \approx 9.2418 a^{3} \end{matrix}$
33	Pentagonal gyrocupolarotunda	25	50	27	$C_{5 v}$ of order 10	$\begin{matrix} A & = (5 + \frac{15}{4} \sqrt{3} + \frac{7}{4} \sqrt{25 + 10 \sqrt{5}}) a^{2} \\ \approx 23.5385 a^{2} \\ V & = \frac{5}{12} (11 + 5 \sqrt{5}) a^{3} \\ \approx 9.2418 a^{3} \end{matrix}$
34	Pentagonal orthobirotunda	30	60	32	$D_{5 h}$ of order 20	$\begin{matrix} A & = ((5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 29.306 a^{2} \\ V & = \frac{1}{6} (45 + 17 \sqrt{5}) a^{3} \\ \approx 13.8355 a^{3} \end{matrix}$
35	Elongated triangular orthobicupola	18	36	20	$D_{3 h}$ of order 12	$\begin{matrix} A & = 2 (6 + \sqrt{3}) a^{2} \\ \approx 15.4641 a^{2} \\ V & = (\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}) a^{3} \\ \approx 4.9551 a^{3} \end{matrix}$
36	Elongated triangular gyrobicupola	18	36	20	$D_{3 d}$ of order 12	$\begin{matrix} A & = 2 (6 + \sqrt{3}) a^{2} \\ \approx 15.4641 a^{2} \\ V & = (\frac{5 \sqrt{2}}{3} + \frac{3 \sqrt{3}}{2}) a^{3} \\ \approx 4.9551 a^{3} \end{matrix}$
37	Elongated square gyrobicupola	24	48	26	$D_{4 d}$ of order 16	$\begin{matrix} A & = 2 (9 + \sqrt{3}) a^{2} \\ \approx 21.4641 a^{2} \\ V & = (4 + \frac{10 \sqrt{2}}{3}) a^{3} \\ \approx 8.714 a^{3} \end{matrix}$
38	Elongated pentagonal orthobicupola	30	60	32	$D_{5 h}$ of order 20	$\begin{matrix} A & = (20 + \sqrt{\frac{5}{2} (10 + \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 27.7711 a^{2} \\ V & = \frac{1}{6} (10 + 8 \sqrt{5} + 15 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 12.3423 a^{3} \end{matrix}$
39	Elongated pentagonal gyrobicupola	30	60	32	$D_{5 d}$ of order 20	$\begin{matrix} A & = (20 + \sqrt{\frac{5}{2} (10 + \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 27.7711 a^{2} \\ V & = \frac{1}{6} (10 + 8 \sqrt{5} + 15 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 12.3423 a^{3} \end{matrix}$
40	Elongated pentagonal orthocupolarotunda	35	70	37	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (60 + \sqrt{10 (190 + 49 \sqrt{5} + 21 \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 33.5385 a^{2} \\ V & = \frac{5}{12} (11 + 5 \sqrt{5} + 6 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 16.936 a^{3} \end{matrix}$
41	Elongated pentagonal gyrocupolarotunda	35	70	37	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (60 + \sqrt{10 (190 + 49 \sqrt{5} + 21 \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 33.5385 a^{2} \\ V & = \frac{5}{12} (11 + 5 \sqrt{5} + 6 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 16.936 a^{3} \end{matrix}$
42	Elongated pentagonal orthobirotunda	40	80	42	$D_{5 h}$ of order 20	$\begin{matrix} A & = (10 + \sqrt{30 (10 + 3 \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 39.306 a^{2} \\ V & = \frac{1}{6} (45 + 17 \sqrt{5} + 15 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 21.5297 a^{3} \end{matrix}$
43	Elongated pentagonal gyrobirotunda	40	80	42	$D_{5 d}$ of order 20	$\begin{matrix} A & = (10 + \sqrt{30 (10 + 3 \sqrt{5} + \sqrt{75 + 30 \sqrt{5}})}) a^{2} \\ \approx 39.306 a^{2} \\ V & = \frac{1}{6} (45 + 17 \sqrt{5} + 15 \sqrt{5 + 2 \sqrt{5}}) a^{3} \\ \approx 21.5297 a^{3} \end{matrix}$
44	Gyroelongated triangular bicupola	18	42	26	$D_{3}$ of order 6	$\begin{matrix} A & = (6 + 5 \sqrt{3}) a^{2} \\ \approx 14.6603 a^{2} \\ V & = \sqrt{2} (\frac{5}{3} + \sqrt{1 + \sqrt{3}}) a^{3} \\ \approx 4.6946 a^{3} \end{matrix}$
45	Gyroelongated square bicupola	24	56	34	$D_{4}$ of order 8	$\begin{matrix} A & = (10 + 6 \sqrt{3}) a^{2} \\ \approx 20.3923 a^{2} \\ V & = (2 + \frac{4}{3} \sqrt{2} + \frac{2}{3} \sqrt{4 + 2 \sqrt{2} + 2 \sqrt{146 + 103 \sqrt{2}}}) a^{3} \\ \approx 8.1536 a^{3} \end{matrix}$
46	Gyroelongated pentagonal bicupola	30	70	42	$D_{5}$ of order 10	$\begin{matrix} A & = \frac{1}{2} (20 + 15 \sqrt{3} + \sqrt{25 + 10 \sqrt{5}}) a^{2} \\ \approx 26.4313 a^{2} \\ V & = (\frac{5}{3} + \frac{4}{3} \sqrt{5} + \frac{5}{6} \sqrt{2 \sqrt{650 + 290 \sqrt{5}} - 2 \sqrt{5} - 2}) a^{3} \\ \approx 11.3974 a^{3} \end{matrix}$
47	Gyroelongated pentagonal cupolarotunda	35	80	47	$C_{5}$ of order 5	$\begin{matrix} A & = \frac{1}{4} (20 + 35 \sqrt{3} + 7 \sqrt{25 + 10 \sqrt{5}}) a^{2} \\ \approx 32.1988 a^{2} \\ V & = (\frac{55}{12} + \frac{25}{12} \sqrt{5} + \frac{5}{6} \sqrt{2 \sqrt{650 + 290 \sqrt{5}} - 2 \sqrt{5} - 2}) a^{3} \\ \approx 15.9911 a^{3} \end{matrix}$
48	Gyroelongated pentagonal birotunda	40	90	52	$D_{5}$ of order 10	$\begin{matrix} A & = (10 \sqrt{3} + 3 \sqrt{25 + 10 \sqrt{5}}) a^{2} \\ \approx 37.9662 a^{2} \\ V & = (\frac{45}{6} + \frac{17}{6} \sqrt{5} + \frac{5}{6} \sqrt{2 \sqrt{650 + 290 \sqrt{5}} - 2 \sqrt{5} - 2}) a^{3} \\ \approx 20.5848 a^{3} \end{matrix}$
49	Augmented triangular prism	7	13	8	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{1}{2} (4 + 3 \sqrt{3}) a^{2} \\ \approx 4.5981 a^{2} \\ V & = \frac{1}{12} (2 \sqrt{2} + 3 \sqrt{3}) a^{3} \\ \approx 0.6687 a^{3} \end{matrix}$
50	Biaugmented triangular prism	8	17	11	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{1}{2} (2 + 5 \sqrt{3}) a^{2} \\ \approx 5.3301 a^{2} \\ V & = \sqrt{\frac{59}{144} + \frac{1}{\sqrt{6}}} a^{3} \\ \approx 0.9044 a^{3} \end{matrix}$
51	Triaugmented triangular prism	9	21	14	$D_{3 h}$ of order 12	$\begin{matrix} A & = \frac{7 \sqrt{3}}{2} a^{2} \\ \approx 6.0622 a^{2} \\ V & = \frac{2 \sqrt{2} + \sqrt{3}}{4} a^{3} \\ \approx 1.1401 a^{3} \end{matrix}$
52	Augmented pentagonal prism	11	19	10	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{1}{2} (8 + 2 \sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 9.173 a^{2} \\ V & = \frac{1}{12} \sqrt{233 + 90 \sqrt{5} + 12 \sqrt{50 + 20 \sqrt{5}}} a^{3} \\ \approx 1.9562 a^{3} \end{matrix}$
53	Biaugmented pentagonal prism	12	23	13	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{1}{2} a^{2} (6 + 4 \sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) \\ \approx 9.9051 a^{2} \\ V & = \frac{1}{12} a^{3} \sqrt{257 + 90 \sqrt{5} + 24 \sqrt{50 + 20 \sqrt{5}}} \\ \approx 2.1919 a^{3} \end{matrix}$
54	Augmented hexagonal prism	13	22	11	$C_{2 v}$ of order 4	$\begin{matrix} A & = (5 + 4 \sqrt{3}) a^{2} \\ \approx 11.9282 a^{2} \\ V & = \frac{1}{6} (\sqrt{2} + 9 \sqrt{3}) a^{3} \\ \approx 2.8338 a^{3} \end{matrix}$
55	Parabiaugmented hexagonal prism	14	26	14	$D_{2 h}$ of order 8	$\begin{matrix} A & = (4 + 5 \sqrt{3}) a^{2} \\ \approx 12.6603 a^{2} \\ V & = \frac{1}{6} (2 \sqrt{2} + 9 \sqrt{3}) a^{3} \\ \approx 3.0695 a^{3} \end{matrix}$
56	Metabiaugmented hexagonal prism	14	26	14	$C_{2 v}$ of order 4	$\begin{matrix} A & = (4 + 5 \sqrt{3}) a^{2} \\ \approx 12.6603 a^{2} \\ V & = \frac{1}{6} (2 \sqrt{2} + 9 \sqrt{3}) a^{3} \\ \approx 3.0695 a^{3} \end{matrix}$
57	Triaugmented hexagonal prism	15	30	17	$D_{3 h}$ of order 12	$\begin{matrix} A & = 3 (1 + 2 \sqrt{3}) a^{2} \\ \approx 13.3923 a^{2} \\ V & = (\frac{1}{\sqrt{2}} + \frac{3 \sqrt{3}}{2}) a^{3} \\ \approx 3.3052 a^{3} \end{matrix}$
58	Augmented dodecahedron	21	35	16	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (5 \sqrt{3} + 11 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 21.0903 a^{2} \\ V & = \frac{1}{24} (95 + 43 \sqrt{5}) a^{3} \\ \approx 7.9646 a^{3} \end{matrix}$
59	Parabiaugmented dodecahedron	22	40	20	$D_{5 d}$ of order 20	$\begin{matrix} A & = \frac{5}{2} (\sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 21.5349 a^{2} \\ V & = \frac{1}{6} (25 + 11 \sqrt{5}) a^{3} \\ \approx 8.2661 a^{3} \end{matrix}$
60	Metabiaugmented dodecahedron	22	40	20	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{5}{2} (\sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 21.5349 a^{2} \\ V & = \frac{1}{6} (25 + 11 \sqrt{5}) a^{3} \\ \approx 8.2661 a^{3} \end{matrix}$
61	Triaugmented dodecahedron	23	45	24	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{3}{4} (5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 21.9795 a^{2} \\ V & = \frac{5}{8} (7 + 3 \sqrt{5}) a^{3} \\ \approx 8.5676 a^{3} \end{matrix}$
62	Metabidiminished icosahedron	10	20	12	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{1}{2} (5 \sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 7.7711 a^{2} \\ V & = \frac{1}{6} (5 + 2 \sqrt{5}) a^{3} \\ \approx 1.5787 a^{3} \end{matrix}$
63	Tridiminished icosahedron	9	15	8	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{4} (5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 7.3265 a^{2} \\ V & = (\frac{5}{8} + \frac{7 \sqrt{5}}{24}) a^{3} \\ \approx 1.2772 a^{3} \end{matrix}$
64	Augmented tridiminished icosahedron	10	18	10	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{4} (7 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 8.1925 a^{2} \\ V & = \frac{1}{24} (15 + 2 \sqrt{2} + 7 \sqrt{5}) a^{3} \\ \approx 1.395 a^{3} \end{matrix}$
65	Augmented truncated tetrahedron	15	27	14	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{2} (6 + 13 \sqrt{3}) a^{2} \\ \approx 14.2583 a^{2} \\ V & = \frac{11}{2 \sqrt{2}} a^{3} \\ \approx 3.8891 a^{3} \end{matrix}$
66	Augmented truncated cube	28	48	22	$C_{4 v}$ of order 8	$\begin{matrix} A & = (15 + 10 \sqrt{2} + 3 \sqrt{3}) a^{2} \\ \approx 34.3383 a^{2} \\ V & = (8 + \frac{16 \sqrt{2}}{3}) a^{3} \\ \approx 15.5425 a^{3} \end{matrix}$
67	Biaugmented truncated cube	32	60	30	$D_{4 h}$ of order 16	$\begin{matrix} A & = 2 (9 + 4 \sqrt{2} + 2 \sqrt{3}) a^{2} \\ \approx 36.2419 a^{2} \\ V & = (9 + 6 \sqrt{2}) a^{3} \\ \approx 17.4853 a^{3} \end{matrix}$
68	Augmented truncated dodecahedron	65	105	42	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (20 + 25 \sqrt{3} + 110 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 102.1821 a^{2} \\ V & = (\frac{505}{12} + \frac{81 \sqrt{5}}{4}) a^{3} \\ \approx 87.3637 a^{3} \end{matrix}$
69	Parabiaugmented truncated dodecahedron	70	120	52	$D_{5 d}$ of order 20	$\begin{matrix} A & = \frac{1}{2} (20 + 15 \sqrt{3} + 50 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 103.3734 a^{2} \\ V & = \frac{1}{12} (515 + 251 \sqrt{5}) a^{3} \\ \approx 89.6878 a^{3} \end{matrix}$
70	Metabiaugmented truncated dodecahedron	70	120	52	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{1}{2} (20 + 15 \sqrt{3} + 50 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 103.3734 a^{2} \\ V & = \frac{1}{12} (515 + 251 \sqrt{5}) a^{3} \\ \approx 89.6878 a^{3} \end{matrix}$
71	Triaugmented truncated dodecahedron	75	135	62	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{4} (60 + 35 \sqrt{3} + 90 \sqrt{5 + 2 \sqrt{5}} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 104.5648 a^{2} \\ V & = \frac{7}{12} (75 + 37 \sqrt{5}) a^{3} \\ \approx 92.0118 a^{3} \end{matrix}$
72	Gyrate rhombicosidodecahedron	60	120	62	$C_{5 v}$ of order 10	$\begin{matrix} A & = (30 + 5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 59.306 a^{2} \\ V & = (20 + \frac{29 \sqrt{5}}{3}) a^{3} \\ \approx 41.6153 a^{3} \end{matrix}$
73	Parabigyrate rhombicosidodecahedron	60	120	62	$D_{5 d}$ of order 20	$\begin{matrix} A & = (30 + 5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 59.306 a^{2} \\ V & = (20 + \frac{29 \sqrt{5}}{3}) a^{3} \\ \approx 41.6153 a^{3} \end{matrix}$
74	Metabigyrate rhombicosidodecahedron	60	120	62	$C_{2 v}$ of order 4	$\begin{matrix} A & = (30 + 5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 59.306 a^{2} \\ V & = (20 + \frac{29 \sqrt{5}}{3}) a^{3} \\ \approx 41.6153 a^{3} \end{matrix}$
75	Trigyrate rhombicosidodecahedron	60	120	62	$C_{3 v}$ of order 6	$\begin{matrix} A & = (30 + 5 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 59.306 a^{2} \\ V & = (20 + \frac{29 \sqrt{5}}{3}) a^{3} \\ \approx 41.6153 a^{3} \end{matrix}$
76	Diminished rhombicosidodecahedron	55	105	52	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (100 + 15 \sqrt{3} + 10 \sqrt{5 + 2 \sqrt{5}} + 11 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 58.1147 a^{2} \\ V & = (\frac{115}{6} + 9 \sqrt{5}) a^{3} \\ \approx 39.2913 a^{3} \end{matrix}$
77	Paragyrate diminished rhombicosidodecahedron	55	105	52	$C_{5 v}$ of order 10	$\begin{matrix} A & = \frac{1}{4} (100 + 15 \sqrt{3} + 10 \sqrt{5 + 2 \sqrt{5}} + 11 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 58.1147 a^{2} \\ V & = (\frac{115}{6} + 9 \sqrt{5}) a^{3} \\ \approx 39.2913 a^{3} \end{matrix}$
78	Metagyrate diminished rhombicosidodecahedron	55	105	52	$C_{s}$ of order 2	$\begin{matrix} A & = \frac{1}{4} (100 + 15 \sqrt{3} + 10 \sqrt{5 + 2 \sqrt{5}} + 11 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 58.1147 a^{2} \\ V & = (\frac{115}{6} + 9 \sqrt{5}) a^{3} \\ \approx 39.2913 a^{3} \end{matrix}$
79	Bigyrate diminished rhombicosidodecahedron	55	105	52	$C_{s}$ of order 2	$\begin{matrix} A & = \frac{1}{4} (100 + 15 \sqrt{3} + 10 \sqrt{5 + 2 \sqrt{5}} + 11 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 58.1147 a^{2} \\ V & = (\frac{115}{6} + 9 \sqrt{5}) a^{3} \\ \approx 39.2913 a^{3} \end{matrix}$
80	Parabidiminished rhombicosidodecahedron	50	90	42	$D_{5 d}$ of order 20	$\begin{matrix} A & = \frac{5}{2} (8 + \sqrt{3} + 2 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 56.9233 a^{2} \\ V & = \frac{5}{3} (11 + 5 \sqrt{5}) a^{3} \\ \approx 36.9672 a^{3} \end{matrix}$
81	Metabidiminished rhombicosidodecahedron	50	90	42	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{5}{2} (8 + \sqrt{3} + 2 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 56.9233 a^{2} \\ V & = \frac{5}{3} (11 + 5 \sqrt{5}) a^{3} \\ \approx 36.9672 a^{3} \end{matrix}$
82	Gyrate bidiminished rhombicosidodecahedron	50	90	42	$C_{s}$ of order 2	$\begin{matrix} A & = \frac{5}{2} (8 + \sqrt{3} + 2 \sqrt{5 + 2 \sqrt{5}} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 56.9233 a^{2} \\ V & = \frac{5}{3} (11 + 5 \sqrt{5}) a^{3} \\ \approx 36.9672 a^{3} \end{matrix}$
83	Tridiminished rhombicosidodecahedron	45	75	32	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{4} (60 + 5 \sqrt{3} + 30 \sqrt{5 + 2 \sqrt{5}} + 9 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 55.732 a^{2} \\ V & = (\frac{35}{2} + \frac{23 \sqrt{5}}{3}) a^{3} \\ \approx 34.6432 a^{3} \end{matrix}$
84	Snub disphenoid	8	18	12	$D_{2 d}$ of order 8	$\begin{matrix} A & = 3 \sqrt{3} a^{2} \\ \approx 5.1962 a^{2} \\ V & \approx 0.8595 a^{3} \end{matrix}$
85	Snub square antiprism	16	40	26	$D_{4 d}$ of order 16	$\begin{matrix} A & = 2 (1 + 3 \sqrt{3}) a^{2} \\ \approx 12.3923 a^{2} \\ V & \approx 3.6012 a^{3} \end{matrix}$
86	Sphenocorona	10	22	14	$C_{2 v}$ of order 4	$\begin{matrix} A & = (2 + 3 \sqrt{3}) a^{2} \\ \approx 7.1962 a^{2} \\ V & = \frac{1}{2} a^{3} \sqrt{1 + 3 \sqrt{\frac{3}{2}} + \sqrt{13 + 3 \sqrt{6}}} \\ \approx 1.5154 a^{3} \end{matrix}$
87	Augmented sphenocorona	11	26	17	$C_{s}$ of order 2	$\begin{matrix} A & = (1 + 4 \sqrt{3}) a^{2} \\ \approx 7.9282 a^{2} \\ V & = \frac{1}{2} a^{3} \sqrt{1 + 3 \sqrt{\frac{3}{2}} + \sqrt{13 + 3 \sqrt{6}}} + \frac{1}{3 \sqrt{2}} \\ \approx 1.7511 a^{3} \end{matrix}$
88	Sphenomegacorona	12	28	18	$C_{2 v}$ of order 4	$\begin{matrix} A & = 2 (1 + 2 \sqrt{3}) a^{2} \\ \approx 8.9282 a^{2} \\ V & \approx 1.9481 a^{3} \end{matrix}$
89	Hebesphenomegacorona	14	33	21	$C_{2 v}$ of order 4	$\begin{matrix} A & = \frac{3}{2} (2 + 3 \sqrt{3}) a^{2} \\ \approx 10.7942 a^{2} \\ V & \approx 2.9129 a^{3} \end{matrix}$
90	Disphenocingulum	16	38	24	$D_{2 d}$ of order 8	$\begin{matrix} A & = (4 + 5 \sqrt{3}) a^{2} \\ \approx 12.6603 a^{2} \\ V & \approx 3.7776 a^{3} \end{matrix}$
91	Bilunabirotunda	14	26	14	$D_{2 h}$ of order 8	$\begin{matrix} A & = (2 + 2 \sqrt{3} + \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 12.346 a^{2} \\ V & = \frac{1}{12} (17 + 9 \sqrt{5}) a^{3} \\ \approx 3.0937 a^{3} \end{matrix}$
92	Triangular hebesphenorotunda	18	36	20	$C_{3 v}$ of order 6	$\begin{matrix} A & = \frac{1}{4} (12 + 19 \sqrt{3} + 3 \sqrt{5 (5 + 2 \sqrt{5})}) a^{2} \\ \approx 16.3887 a^{2} \\ V & = (\frac{5}{2} + \frac{7 \sqrt{5}}{6}) a^{3} \\ \approx 5.1087 a^{3} \end{matrix}$